Resolvents of the adjacency matrix also are commonly used to determine properties of nodes
in a single-layer network; see, e.g., Estrada and Higham [1]} and Katz [2]}. We
define the modified tensor resolvent,
\({\rm res}_0(\mathcal {A},\alpha )=(I-\alpha \mathcal {A})^{-1}-I= \sum _{p=1}^{\infty }\alpha ^p\mathcal {A}^p,\) 
